Cube Roots of Unity and the Equation
The question asks for the solutions (\(z\)) to the equation \((z-100)^3 + 1000 = 0\). We are given that \(\omega \ne 1\) is a cube root of unity.
Key properties of the cube roots of unity (\(1, \omega, \omega^2\)) are:
- The cube roots satisfy the equation \(x^3 = 1\).
- They follow the rules: \(\omega^3 = 1\) and \(1 + \omega + \omega^2 = 0\).
The value \(\omega\) represents one of the complex cube roots of unity, often \(e^{i 2\pi/3}\) or \(e^{i 4\pi/3}\).
Solving the Cubic Equation
First, we rearrange the given equation to isolate the cubic term:
\((z-100)^3 + 1000 = 0\)
Subtract 1000 from both sides:
\((z-100)^3 = -1000\)
To make the equation easier to work with, let's use a substitution. Let \(X = z - 100\). The equation now becomes:
\(X^3 = -1000\)
Finding the Cube Roots of -1000
We need to find the three values of \(X\) that satisfy \(X^3 = -1000\). We know that \((-10)^3 = -1000\), so one solution for \(X\) is \(X = -10\).
The three cube roots of a number \(C\) can be expressed using the principal cube root and the cube roots of unity. If \(\sqrt[3]{C}\) is the principal cube root, the three roots are \(\sqrt[3]{C}\), \(\sqrt[3]{C}\omega\), and \(\sqrt[3]{C}\omega^2\).
In this case, \(C = -1000\), and the principal cube root is \(-10\). Therefore, the three possible values for \(X\) are:
- \(X_1 = -10\)
- \(X_2 = -10 \times \omega = -10\omega\)
- \(X_3 = -10 \times \omega^2 = -10\omega^2\)
Deriving the Solutions for z
Now, we need to find the values of \(z\) by substituting back \(X = z - 100\) for each of the three possible values of \(X\) found above.
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Case 1: \(X_1 = -10\)
\(z - 100 = -10\)
To find \(z\), add 100 to both sides:
\(z = 100 - 10\)
\(z = 90\)
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Case 2: \(X_2 = -10\omega\)
\(z - 100 = -10\omega\)
Add 100 to both sides:
\(z = 100 - 10\omega\)
We can factor out 10 from the terms on the right side:
\(z = 10(10 - \omega)\)
-
Case 3: \(X_3 = -10\omega^2\)
\(z - 100 = -10\omega^2\)
Add 100 to both sides:
\(z = 100 - 10\omega^2\)
Factor out 10:
\(z = 10(10 - \omega^2)\)
Combining the results from the three cases, the solutions (\(z\)) for the equation \((z-100)^3 + 1000 = 0\) are \(90\), \(10(10 - \omega)\), and \(10(10 - \omega^2)\).
Similar Questions
- If \(z\) is any complex number and \(iz^3 + z^2 - z + i = 0\), where \(i = \sqrt{-1}\), then what is the value of \((|z|+1)^2\)?
- Let \(z_1\) and \(z_2\) be two complex numbers such that \(\left|\frac{z_1+z_2}{z_1-z_2}\right| = 1\), then what is \(\text{Re}\left(\frac{z_1}{z_2}\right)+1\) equal to ?
- What is \((1+i)^4 + (1-i)^4\) equal to, where \(i = \sqrt{-1}\)?
- If \(\omega \neq 1\) is a cube root of unity, then what is \((1 + \omega - \omega^2)^{100} + (1 - \omega + \omega^2)^{100}\) equal to?
- What is the value of the sum
\(\sum_{n=1}^{20}(i^{n-1} + i^n + i^{n+1})\)
where \(i = \sqrt{-1}\)? - If \(x, y\) and \(z\) are the cube roots of unity, then what is the value of \(xy + yz + zx\)?
- If \(\alpha\), \(\beta\), \(\gamma\) are cube roots of -8, then what is \(\frac{\alpha^2 p^2 + \beta^2 q^2 + \gamma^2 r^2}{\beta^2 p^2 + \gamma^2 q^2 + \alpha^2 r^2}\) equal to ?
- What is the value of \(\left|\frac{Z_1}{Z_2}\right|\)?
- What is the value of \(\frac{1}{2} + \text{Re}\left(\frac{Z_1}{Z_2}\right)\)?
- Consider the following inequalities :
I. \(1 + 4i > 3 + 2i\)
II. \(2 + 3i < 3 + 4i\)
III. \(4 + 3i > 3 + 4i\)
where \(i = \sqrt{-1}\)
How many of the above are valid ?
Important Questions from Complex Numbers
If \(x + iy = \sqrt {\frac{{a + ib}}{{c + id}}}\), then the value of x2 + y2 is -
If x = cos θ + i sin θ, then the value of \({x^n} + \frac{1}{{{x^n}}}\) is:
If \(\left| {\begin{array}{*{20}{c}} {6i}&{ - 3i}&1\\ 4&{3i}&{ - 1}\\ {20}&3&i \end{array}} \right| = x + iy\), then the values of x and y are:
If iz3 + z2 - z + i = 0, then the value of |z| is:
Nature of the triangle formed by the points representing the complex numbers 3 + 4i, 8 - 6i and 13 + 9i is:
